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Let $X_1,~X_2,\cdots$ be a sequence of i.i.d random variables which are supposed to be observed in sequence. The $n$th value in the sequence is a $k-record~value$ if exactly $k$ of the first $n$ values (including $X_n$) are at least as large as it. Let ${\bf R}_k$ denote the ordered set of $k$-record values. The famous Ignatov's Theorem states that the random sets ${\bf R}_k(k=1,2,\cdots)$ are independent with common distribution. We introduce one new record named $recent-k-record$ (RkR in short) in this paper: $X_n$ is a $j$-RkR if there are exactly $j$ values at least as large as $X_n$ in $X_{n-k},~X_{n-k+1},\cdots,~X_{n-1}$. It turns out that RkR brings many interesting problems and some novel properties such as prediction rule and Poisson approximation which are proved in this paper. One application named "No Good Record" via the Lov{\'a}sz Local Lemma is also provided. We conclude this paper with some possible connection with scan statistics.

To approximate the trajectories of a stochastic process by the solution of some differential equation is widely used in the fields of probability, computer science and combinatorics. In this paper, the convergence of coupon collecting processes is studied via the differential equation techniques originally proposed in Wormald(1995) and modified in Warnke(2019). In other words, we give a novel approach to analysis the classical coupon collector's problem and keep track of all coupons in the process of collecting.

In this paper, the parameter estimation of ARMA(p,q) model is given by approximate Bayesian computation algorithm. In order to improve the sampling efficiency of the algorithm, approximate Bayesian computation should select as many statistics as possible with parameter information in low dimension. Firstly, we use the autocorrelation coefficient of the first p+q order sample as the statistic and obtain an approximate Bayesian estimation of the AR coefficient, transforming the ARMA(p,q) model into the MA(q) model. Considering the first q order sample autocorrelation functions and sample variance as the statistics, the approximate Bayesian estimation of MA coefficient and white noise variances can be given. The method mentioned above is more accurate and powerful than the maximum likelihood estimation, which is verified by the numerical simulations and experiment study.

In this paper, we investigate the induced subgraphs of percolated random geometric graphs, and get some asymptotic results for the expected number of the subgraphs. Moreover, we get the Poisson approximation for the counting by Stein's method. We also present some similar results for the expectation of Betti number of the associated percolated random geometric complexes., Comment: 15pages